aa r X i v : . [ m a t h . M G ] J u l Generalized Busemann inequality
Alexander E. Litvak and Dmitry Zaporozhets ∗ Abstract
We present a result which simultaneously extends the Busemann inter-section inequality to the case of non-integer moments of the correspondingvolumes and the Busemann random simplex inequality to the case of sim-plices of smaller dimensions.
AMS 2010 Classification: primary: 60D05, 52A22; secondary: 52A23, 46B06
Keywords:
Blaschke–Petkantschin formula, Busemann intersection inequality,Busemann random simplex inequality, convex hull, expected volume, Furstenberg–Tzkoni formula, random section.
We start with basic notation of integral geometry following [15]. Let d ≥ R d with non-empty interior is called a convexbody. The unit Euclidean ball in R k is denoted by B k . By | · | we denote the d -dimensional volume. Given k ≤ d , slightly abusing notation, considering the setsintersected with k -dimensional affine subspaces or the convex hulls of k + 1 points,we denote the k -dimensional volume by | · | as well.For p > κ p := π p/ Γ (cid:0) p + 1 (cid:1) and ω p = pκ p . (1) ∗ The work of this author was supported by the Foundation for the Advancement of TheoreticalPhysics and Mathematics “BASIS”. k one has κ k = | B k | and ω k = | ∂ B k | . We will also need thefollowing numbers b q,k := ω q − k +1 · · · ω q ω · · · ω k . (2)For k ∈ { , . . . , d } , the linear (resp., affine) Grassmannian of k -dimensional linear(resp., affine) subspaces of R d is denoted by G d,k (resp., A d,k ) and is equipped witha unique rotation invariant (resp., rigid motion invariant) Haar measure ν d,k (resp., µ d,k ), normalized by ν d,k ( G d,k ) = 1 and µ d,k (cid:0)(cid:8) E ∈ A d,k : E ∩ B d = ∅ (cid:9)(cid:1) = κ d − k , respectively. For L ∈ G d,k (resp., E ∈ A d,k ) we denote by λ L (resp., λ E ) the k -dimensional Lebesgue measures on L (resp., E ). The seminal Busemann intersection inequality [1] states that for any convex body K ⊂ R d and for k = d − Z G d,k | K ∩ L | d ν d,k (d L ) ≤ κ dk κ kd | K | k . (3)This inequality was later generalized in [3, 10] for all k = 1 , . . . , d −
1. Using thepolar coordinates, it is easy to see that for k = 1 the inequality turns to the equality.Moreover, the equality in the case k = 1 can be generalized to other moments asfollows: Z G d, | K ∩ L | d + p ν d,k (d L ) = ( d + p )2 d + p dκ d Z K k | x | p d x , p ≥ − d + k + 1 . (4)If k ≥ K is an ellipsoid centered at the origin,and in this case (3) turns to the classical Fustenberg–Tskoni formula [7].The affine counterpart of (3) was obtained by Schneider [14], namely Z A d,k | K ∩ E | d +1 µ d,k (d E ) ≤ κ d +1 k κ d ( k +1) κ k +1 d κ k ( d +1) | K | k +1 . (5)2s above, for k = 1 the inequality turns to equality, although it is not as trivialas in the linear case, see [5] for d = 2 and [12] for any d . As in the linear case,this equality can be generalized to other moments. It was done independently in [4,Eq. (21)] and [13, Eq. (34)]: Z A d, | K ∩ E | p + d +1 µ d, ( dE ) = ( d + p ) ( d + p + 1)2 dκ d Z K | x − x | p d x d x . If k ≥
2, then equality holds if and only if K is an ellipsoid.Gardner [8] generalized (3) and (5) to bounded Borel sets and characterizedthe equality cases. Recently Dann, Paouris, and Pivovarov [6] extended (3), (5) tobounded integrable functions.Given that (4) and (5) for k = 1 (which are the equalities in this case) can begeneralized to other moments, the following question arises naturally. Question I.
Is it possible to generalize (3) and (5) to the case of non-integer mo-ments of | K ∩ L | ? Another group of inequalities deals with the volume of the random simplex in abody. The classical Busemann random simplex inequality states that | K | d +1 ≤ ( d + 1)! κ d +1 d κ d − d +1 Z K d | conv(0 , x , . . . , x d ) | d x . . . d x d . This inequality can be generalized (see, e.g., [15, Theorem 8.6.1.]) as follows: forevery p ≥ | K | p + d ≤ ( d !) p κ p + dd κ dd + p b d + p,d Z K d | conv(0 , x , . . . , x d ) | p d x . . . d x d . (6)The equality holds if and only if K is a centered ellipsoid.The affine counterpart of (6) is known as the Blaschke-Gr¨omer inequality [11]:for every p ≥ | K | p + d +1 ≤ ( d !) p b d + p,d κ p + d +1 d κ d +1 d + p κ ( d +1)( d + p ) κ d ( d + p +1) Z K d +1 | conv( x , . . . , x d ) | p d x . . . d x d . (7)3s before, the equality holds if and only if K is an ellipsoid.The following question arises naturally. Question II.
Is it possible to generalize (6) and (7) to the case of random simplicesof all dimensions k = 1 , . . . , d ? The aim of this note is to positively answer Questions I, II presenting inequali-ties that generalizes both the Busemann intersection inequality and the Busemannrandom simplex inequality.
Our first theorem generalizes (3) and (6).
Theorem 2.1.
For any convex body K ⊂ R d , k ∈ { , , . . . , d } , and any real number p ≥ − d + k + 1 , Z G d,k | K ∩ L | p + d ν d,k (d L ) ≤ ( k !) p κ d + pk κ kd + p b d + p,k b d,k Z K k | conv(0 , x , . . . , x k ) | p d x . . . d x k . (8) For k ≥ the equality holds if and only if K is a non-degenerate ellipsoid centeredat the origin. Remarks.
1. Applying (8) with p = 0 we obtain (3), while applying it with k = d we obtain(6).2. It was shown in [9, Theorem 1.6] that if K is a non-degenerate ellipsoid cen-tered at the origin, then one has the equality in (8).3. In the probabilistic language it may be formulated as E | K ∩ ξ | p + d ≤ ( k !) p κ d + pk κ kd + p b d + p,k b d,k | K | k E | conv(0 , X , . . . , X k ) | p where X , . . . , X k are i.i.d. copies of a random variable uniformly distributedin K and ξ is uniformly distributed in G d,k .Our second theorem generalizes (5) and (7).4 heorem 2.2. For any convex body K ⊂ R d , k ∈ { , , . . . , d } , and any real number p ≥ − d + k + 1 , Z A d,k | K ∩ E | p + d +1 µ d,k ( dE ) ≤ C ( k, p, d ) Z K k +1 | conv( x , . . . , x k ) | p d x . . . d x k , (9) where C ( k, p, d ) = ( k !) p κ p + d +1 k κ k +1 d + p κ ( k +1)( d + p ) κ k ( d + p )+ k b d + p,k b d,k . For k ≥ the equality holds if and only if K is a non-degenerate ellipsoid. Remarks.
1. Applying (9) with p = 0 we obtain (5), while applying it with k = d we obtain(7).2. It was shown in [9, Theorem 1.4] that if K is a non-degenerate ellipsoid, thenone has the equality in (9).3. In probabilistic language it may be formulated as E | K ∩ η | p + d +1 ≤ C ′ ( k, p, d ) | K | k +1 V d − k ( K ) E | conv( X , X , . . . , X k ) | p , where C ′ ( k, p, d ) = d ! ( k !) p − ( d − k )! κ d κ d − k κ p + dk κ k +1 d + p κ ( k +1)( d + p ) κ k ( d + p )+ k b d + p,k b d,k ,X , X , . . . , X k are i.i.d. copies of a random variable uniformly distributedin K , η is uniformly distributed among all affine k -planes intersected K , and V d − k is the ( d − k )-th intrinsic volume of K defined by the Crofton formula [15,Theorem 5.1.1] as the normalized measure of all affine k -planes intersected K : V d − k := (cid:18) dk (cid:19) κ d κ k κ d − k µ d,k ( { E ∈ A d,k : E ∩ K = ∅} ) . Proofs
Recall that b d,k is defined by (2). Given points x , x , . . . , x k ∈ R d we denote V k = V ( x , x , . . . , x k ) := | conv( x , x , . . . , x k ) | and V ,k = V ( x , . . . , x k ) := | conv(0 , x , . . . , x k ) | . In our further calculations we will need to integrate some non-negative measurablefunction h of k -tuples of points in R d . To this end, we first integrate over the k -tuples of points in a fixed k -dimensional linear subspace L with respect to theproduct measure λ kL and then we integrate over G d,k with respect to ν d,k . Thecorresponding transformation formula is known as the linear Blaschke–Petkantschinformula (see [15, Theorem 7.2.1]): Z ( R d ) k h d x . . . d x k = ( k !) d − k b d,k Z G d,k Z L k h V d − k ,k λ L (d x ) . . . λ L (d x k ) ν d,k (d L ) , (10)where h = h ( x , . . . , x k ). The following is an affine counterpart of (10), Z ( R d ) k +1 h d x . . . d x k = ( k !) d − k b d,k Z A d,k Z E k +1 h V d − kk λ E (d x ) . . . λ E (d x k ) µ d,k (d E ) , (11)where h = h ( x , x , . . . , x k ) (see [15, Theorem 7.2.7]). Let J := Z K k V p ,k d x . . . d x k = Z ( R d ) k V p ,k k Y i =1 K ( x i ) d x . . . d x k . Applying the linear Blaschke–Petkantschin formula (10) with the function h ( x , . . . , x k ) := V p ,k k Y i =1 K ( x i ) ,
6e observe J = ( k !) d − k b d,k Z G d,k Z L k V p + d − k ,k k Y i =1 K ( x i ) λ L (d x ) . . . λ L (d x k ) ν d,k (d L )= ( k !) d − k b d,k Z G d,k Z ( K ∩ L ) k V p + d − k ,k λ L (d x ) . . . λ L (d x k ) ν d,k (d L ) . (12)Fix L ∈ G d,k . Applying (6) with p + d − k and k instead of p and d , we obtain( k !) p + d − k κ d + pk κ kd + p b d + p,k Z ( K ∩ L ) k V p + d − k ,k λ L (d x ) . . . λ L (d x k ) ≥ | K ∩ L | p + d , (13)which together with (12) implies (8).Finally we consider the equality case. As was mentioned above, the equality holdsfor ellipsoids, see [9, Theorem 1.6]. Conversely, suppose that (8) turns to equality.Then it follows from (12) that (13) turns to equality for almost all L ∈ G d,k which,in fact, means that it is true for all L ∈ G d,k . Indeed, if for some L ∈ G d,k wehad a strict inequality in (13), then the same would be true for some neighborhoodof L which would contradict to the fact that (13) turns to equality for almost all L ∈ G d,k . Thus, according to the equality case in (6), K ∩ L is a centered ellipsoidfor all L ∈ G d,k . Now it remains to apply the following lemma from [2, (16.12)]: iffor any E ∈ A d,k passing through some fixed point from the interior of K the inter-section K ∩ E happens to be a k -dimensional ellipsoid, then K is an ellipsoid itself. ✷ The proof is similar to the previous one. Let J := Z K k +1 V pk d x . . . d x k = Z ( R d ) k +1 V pk k Y i =0 E ( x i ) d x . . . d x k . Applying the affine Blaschke–Petkantschin formula (11) with the function h ( x , . . . , x k ) := | conv( x , . . . , x k ) | p k Y i =0 E ( x i ) ,
7e observe J = ( k !) d − k b d,k Z A d,k Z E k +1 V p + d − kk k Y i =0 K ( x i ) λ E (d x ) . . . λ E (d x k ) µ d,k (d E )= ( k !) d − k b d,k Z A d,k Z ( K ∩ E ) k +1 V p + d − kk λ E (d x ) . . . λ E (d x k ) µ d,k (d E ) . (14)Fix E ∈ A d,k . Applying (7) with p + d − k and k instead of p and d , we obtain( k !) d − k + p b d + p,k κ p + d +1 k κ k +1 d + p κ ( k +1)( d + p ) κ k ( d + p +1) Z ( K ∩ E ) k +1 V p + d − kk λ E (d x ) . . . λ E (d x k ) ≥ | K ∩ E | d + p +1 which together with (14) implies (9).The equality case is treated the same way as in the linear case. ✷ References [1] H. Busemann,
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Probability and itsApplications (New York), Springer-Verlag, Berlin, 2008Alexander E. LitvakDept. of Math. and Stat. Sciences,University of Alberta,Edmonton, AB, Canada, T6G 2G1. e-mail: [email protected]
Dmitry ZaporozhetsSt. Petersburg Department ofSteklov Institute of MathematicsSt. Petersburg, Russia e-mail: [email protected]: [email protected]